Slide rule



J. R. BLAND June 17, 1947.

SLIDE RULE Filed oct.' 1'?, 1944 s nm.-

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Patented June 17, 1947 `2,422,649 SLIDE RULE James a. Bland, Eascport, Md., saumur to Kennel & Esser Company, Hoboken, N. J., a corporation of New Jersey Application October 17, 1944, Serial No. 559,020

(Cl. 23S- 70) 6 Claims.

1 This invention relates to slide rules and has as its object to provide a rule of enhanced power and in the use of which by a process requiring lgenerally one but no more than one movement either of the hair-line or of the slide for each number in the expression to be computed. The improved capabilities of the new rule can be best explained with reference to an actual embodiment such as is shown by way of illustration in the accompanying drawing in which Figure 1'is a view of one face of a slide rule in accordance with the invention and Figure 2 is a view of the other face.

As a matter of convenient reference, the face of the rule shown in Figure 1 will hereinafter be considered to be the front face and the face shown in Figure 2 to be the rear face. I have chosen to show the invention as applied to a rule which is the same in mechanical structure as that shown in Patent Nc. 2,170,144 granted August 22, 1939. As in the said patent, reference letters H and J designate parallel bars secured together by end plates so as to provide a slotted body in the slot of which a slide N is relatively movable. In the said patent, the reference letter X designates a runner having transparent faces carrying a hair' line Y. For the sake of clarity, the runner has been omitted from the present drawings.

The slide is?, Figure 1, is provided with the usual C scale, graduated in accordance with the logarithme` of numbers from 1 to l0, and of unit length. The body carries the usual D scale which is exactly the as the C scaie and these two scales are herein considered to be basic scaies since all other scales are directly associated with them so that all slide rule operations are most easily explainabie by means of them.

Above the C seais, as viewed in Figure l, is a CT scale which'is identical anden-extensive with the C scaie except that it is inverted with respect thereto. The term co-extensive as used herein includes 'the meaning of registry, i. e., the im clusicn Within common parallel terminal lines, insofar as effective extent is concerned.

Above the CI scale is the CIF scale which is the same as the C1 scale except that it is folded at 1r, while the immediately adjacent CF scale is identical with the C scale except that it is folded at vr. The immediately adjacent DF scale, on the body, is identical with the CF scale.

On its rear face, the slide carries a. B scale which is co-extensive with the C scale and is graduated in accordance with the logarithms of numbers from 1 to 100. Below the B scale, as viewed in Figure 2, the slide carries the co-extensive trigonometric scales T, ST and S. The T scale is graduated in accordance with the logarithms of the values of the natural tangents from 5 43 to 45, the ST scale is graduated in accordance with the logarithms of the values of 2 the natural sines of the angles from 34' to 5 43", while the S scale is like the ST scale except that its range is from 5 43' to 90.

On its rear face, the body carries K, A, D and L scales all co-extensive with the first-mentioned D scale. The K scale is graduated in accordance,

with the logarithms of numbers from 1 to 1000, the A scale isidentical with the B scale, hereinbefore mentioned, the D scale is identical with the mst-mentioned D scale, and the L scale is uniformly graduatedin ten main primary divisions and appropriate subdivisions.

The LL scale, as here shown, is in three sections, LLI, LL! and LLB each of fullunit length, of which LLI appears on the rear face, and LL! and LL3 appear on the front face. Each section is coextensive with the C and D scales and the entire scale is graduated in accordance with the logarithms of the logarithms of numbers greater than unity, Finally, the rule is provided with an LLO scale in three sections each of full unit length of which section L10I .appears on the rear face and sections LLO2 and LLO3 appear on the iront face. 'I'he sections oi this scale are each cci-extensive with the basic scales and the scale as a whole is graduated in accord ance with the logarithms of the co-logarithms of the positive numbers less than unity or, to ex press it differently, in accordance with the logarithms of the logarithms of the reciprocals of the numbers of the LL scale.

lThe range selected for the LLO scales corresponds to the range used for the LL scales. In

the illustrated embodiment e and c are opposite an index of the D scale. In the case of a rule in which it is necessary to conserve space on the face of the rule, a different range of vaines could be selected for the LL scales and a correrw spending range would be selected for the ILO scales. In such a case, some number other than c would be placed opposite the index ci D. iront face. The choice in this matter would depend upon a selection of the most useful range of values on the LL and on the LLO scales. For example, if a single LL scale section were to be used,

the range of values chosen might be from 1.585

, (approximately) to 100. In this case, the flrst oi these numbers would be set opposite the left index of D, while the would be set opposite the right index of D. With this choice for a single line LL scale, the co-extensive single line LL@ scale would have the range of 0.631 (approxim mately) to 0.01.' The rst of these numbers would be placed opposite the left index of'D, while the point 0.01 would be set opposite the right index of D. A

As here shown, the LLO scale is graduated in accordance with the logarithms of the co-logarithms of positive numbers between e1 equals 0.9900, and er-1 equals 0.9048, eroand e-1 equals 0.3670, and er1 and el0 equals 0.000052, respectively. These graduations are so spaced that, like the LL scale, the trigonometric scales and the K, A, B, L, DF, CF, CIF and CI scales, the LLO scale is directly associated with numbers on the basic scales C and D. By directly associated," I means that when the rule is closed (i. e. when the indices are aligned) and the hairline is pushed to a value on any scale other than the C and D scales, the value of the function of this scale associated with the marked value is directly readable at the hairline on scale C. For example, in the c'ase of the trigonometric scales, if the hairline is pushed to an angle on the S scale, the sine of that angle will be marked at the hairline on the C scale; if the hairline is pushed to a number on the B scale, the square root of the number will be marked at the hairline on the C scale; if the hairline is pushed to a number on the CI scale, the reciprocal of that number will be marked at the hairline on the C scale; and if the hairline is pushed to a number on the CF scale, the hairline will mark on the C scale the number divided by f.

A great advantage arises from the direct association of the LLO scale with the basic scales C and D instead of with the scales A and B, which latter has been the practice heretofore as illustrated. for example, in the above mentioned patent. Every scale outside of the C and D scales is directly associated with the C and D scales. Consequently, the rules governing the placing of the decimal point and the result arrived at in the use of the LLC scale are very similar to those governing the placing of the decimal point in a result arrived at in the use of the LL scales. Operating rules of a Very unlike and therefore confusing nature govern the use of' the two correspending sets of log log scales in prior art rules of the type shown in the said patent.

The new LLC scale gives additional power to the rule and makes the operator more accurate in making settings, since the principles are similar to those involved in other settings, and this simplicity is achieved by the described association of the LLO scale with the basic C and D scales. The operator has the advantage of being able to obtain negative powers of numbers greater than 1. and negative powers of numbers less than 1 simultaneously. The LL and LLO scales may be used in cooperation with the C and D scales in one operation. For example, if the operator wishes to find 4*'2, he sets the left index of C opposite 4 on the LL3 scale, pushes the hairline to 2 on the C scale, and at the hairline reads 0.062 on the LLO3 section.

The, advantages arising from the new association of scales will be evident from the following examples:

Example 1.-Evaluate:

3.4701, 3.47--1, 3.47001, and 3.47-om SolutiovaSet hairline on glass indicator to 3.47 on scale LL3 At hairline on LL2 read 3.47-1=1.1325 At hairline on LLO2 read 3.47-1=0.883 At hairline on LLl read 3.471=1.0125 At hairline on LLOi read 3.471=0.9877

Example 2.-Find the logarithms to the base e (=2.7183 approximately) of the numbers 1.135, and its reciprocal 0.883, 1.0125, and its reciprocal 0.9877.

Solution-Set hairline on glass indicator to 1.135 of scale LLZ At hairline en D scale read loge 1.1325 (found on At hairline on D scale read loge 0.883 (found on LLOZ) =-0.1242

At hairline on D scale read loge 1.0125 (found on LLI)=0.01242 At hairline on D scale read loge 0.98765 (found on LLOI) =0.01242 Observe in Example l, that the answers for like exponents were found on like numbered scales, and in Example 2 that the decimal point in the answers were similarly placed when the answers were derived from like numbered scales. In prior art rules, no such simplicity obtained because the scales marked LLOO and LLO were read against the A and B scales whereas the scales marked LLI, LL2, and LL3 were read against the C and D scales for practically every important problem. It was a matter of common experience, that students had great diiculty in using scales marked LLOO and LLO on prior art rules because of difficulties arising because the A and B scales were repeated scales. These difliculties do not arise in the operation of the slide rule of the present invention, since like rules of operation apply to both sets of the log log scales.

In the use of the new slide rule most practical applications of the log log scales are operated in conjunction with the C and D scales. However many important applications involve their use with other scales. In every case likeness of method of operation and of placing the decimal point obtain when the rule of this invention is employed. The following examples will illustrate this fact.

Example 3.-Evaluate e0m 30 and e-sin 30 Solution.-Using the rule of the present invention, close the rule, that is, set the left mark numbered 1 on scale C opposite the same numbered left mark of scale D, push the hairline on the glass indicator to 30 on scale S, at the hairline on scale LL2 read 1.65 and on scale LLO2 read e-Sin 3""=0.606. Observe that like numbered log log scales LLZ and LLOZ were used.

Example 4.-Evaluate 500570 and 5-00s 70 Solution-Opposite 5 on scale LL3 draw left index of C, push hairline to 70 (red) on scale S, at the hairline on LL2 read 1.7'3i=50s 70 and at the hairline read 5"'0"s 70:0.577 on LLOZ.

Example 5.-Evaluate 0.201025 and 0.2-00125" SoZution.-Opposite 0.2 on scale LLO3, set l of scale C, push hairline to 25 on scale T,

At hairline read 0.2tan 25=0.472 on scale LLOZ At hairline read 0.2*00025=2.12 on scale LL2.

Example 6.--Evaluate 1 L 0.1502 and 0.15 5-2 Solution-Draw index of CI opposite 0.15 on scale LLO3, push hairline to 5.2 on CI,

At hairline read 0.1552=o.694 on scale LL02 At hairline read (0.15) 52'=1.44 on scale LL2 Example 7.--Eva1uate 0.804? and o soi/ Solution- Draw left index of B opposite 0.80

on scale LLOI, push hairline to 5 on left oi' B,

half

blauem- Push the hairline to 33 on the middle K scale, At hairline read e3=-o4o4 on LLos At hairline read Wi-:24.7 on LL3 Example 9.-Evaluate eend-logia and 1mi-logia" Solution-Push the hairline to .624 on scale Lv At hairline read "*1""=o.c56 on scale LL02 At hairline read Example raf-Emme@ 2.111#` and 2.1-v' Solution-Draw right lindex of CF to 2.1 of LL2, push the hairline to 6 of CF At hairline read 2.1/'=4.12 on LL: At hairline read 2.16/'=0.242 on LLOI Example 11,-Evaluate Soluton.-Draw index of CIF opposite .25 of LLOI push hairline to 2 of CIF At hairline read 1 o.25)m=o.11 on LL03 At hairline read l (0.25) '=9.1 on LL The examples given above show the same likeness of operation and of placing the decimal point for the log log scales in conjunction with each of the other scales on the rule. Examples -for the A, D, DF, and ST scales are not given since they would duplicate in principal examples already given.

The slide rule of the presentV invention has more power than the prior rules as regards the log log scales. For example, the expression 0.2"xx 5' occurring in Example 5 could not be evaluated with a. single setting of the slide on the prior art slide rules, neither could such expressions as 2.9-13 and 0.29-0-31. These exressions involve the use of scales LLI, LL2 and LL3 in conjunction with the C scale and the scales LLOI, LLO2, and LLO3 on the rule of this invention and are easily evaluated with a single setting of the slide.

Thus to evaluate 2.9*03 Draw the index of C opposite 2.9 on LLI, Opposite 3 on C read 2.9-3:0726 on LLOZ and to evaluate 0.29-031 amaca@ 6 Draw the index oi' C opposite 0.29 on LLOS Opposite 31 on C read 0.291=1.467 on LL2.

Hyperbolic functions are being used more and more in engineering practice. With the new LLOI, LLO2, and LLO3 scales in conjunction with the scales marked LLI, LL2, and LLI, it is a simple matter to ilnd the value of various hyperbolic functions. Thus to nd sinh 2 and cosh 2, push the hairline to 2 on scale D and read at the hairline on LL3 7.39 and on LLO3 0.135. Hence sin h 2=1/2(7.390.135)= 3.627 and cosh 2=1,(7.39+0.135) :3.762. It is to be observed in this process that sinh 2 and cosh 2 were easily obtained since both the numbers 7.39 and 0.135 were found at once on log log scales LL3 and LLO3 having like numbers when the hairline was pushed to 2 on scale D. With the prior art slide rules, the number 7.39 is found as above and the number 0.135 is found opposite 2 of one of the left A scales on the LLO scale. Instead of a single setting of the hairline on the rule of the present invention, two settings are necessary on the prior art rules and complicated rules of determining what scales apply are involved.

As illustrating the obtaining of a final result by continuous progressive manipulations of the new slide rule in the evaluation of expressions involving logarlthms of numbers less than unity, which evaluation would require resetting in the use of any prior art slide rule, the following examples are given:

Example 12.-Evaluate 17 log, 0.04

Solution-To on LLO3 scale Draw 31 onB (right) scale, Push indicator to 17 on CF scale and Read 9.83 on DF scale.

Example 13.-Evaluate log. 0.7145 sin 45 cos 70 Solution-To 0.7145 on LLO2 scale Draw 70 on S (red), Push indicator to 45 on S scale and Read .6955 on D scale.

Example 14.-Evaluate.

cos 79 05 log., 0.9525 sec 70 10 sin 55 Solution- To 0.9525 on LLOI Draw 55,on S,

Push hairline to 79 05 on S (red),

Draw 70 10 on S (red) to hairline and r Read at the index .03319 on D.

Example 15.-Evaluate 016.2 log, 0.0074 csc 60 Solution-To 0.0074 on LLO3 scale Draw 60 on Sscale, Push hairline to 16.2 on B and Read 22.8 on D scale.

Observe that it is impossible, by means of prior art rules of any kind, to evaluate each of the expressions in Examples 12, 13, 14 and 15 by continuous progressive manipulations, that is by a process requiring generally one but no more than one movement either of the hairline or of the slide for each number in the expression to be computed.

Example 16.-Evaluate e sin 40 {Ii-5 Solution.

Push hairline to 0.72 on scale LLOZ, Draw 40 of S scale under the hairline, Push the hairline to 25 of T scale,

Draw 35 of B (right) under the hairline, Push the hairline to index of C,

At the hairline read 30412 on LLI scale.

Example l 7.-Evalua'te -lg. 1.311 um 40 1 0 g cos 47@ cot 35 Solution.

Push the hairline to 1.311 on scale L LZ, Draw ^-7 oi scale S (red) under the hairline, Push the hairline to on scale T,

Draw 3.7 of scale CIF under the hairline, Push the hairline to 35 of scale T, At the hairline on scale LLO3 read 0.0664.

Ezample 18.--Evaluate nog. mail@ e -sin 40.

Solution.

iush hairline to 0.65 on scale 1,1102, Draw 40 of S scale under the hairline, Push the hairline to 23 on B right,

At the hairline read 0.0401 on scale LLOS.

rlwo resettings would be necessary to perform this evaluation by means of the slide rule of Patent No. 2,170,144,

Another feature of improvement over the above mentioned prior patent lies in the disposition of the sine, co-sine, tangent and co-tangent numbers. In said prior patent, the co-sine and cotangent numbers are in red and are to the right of the markings, whereas the sine and tangent numbers are in black and to the left of the markings, adjacent numbers being inclined toward the common marking. As shown in Figure 2, the cosine and co-tangent numbers, which are in red, are placed at the left of the markings, while the sine and tangent numbers, which are in black, are placed to the left of the markings, and adjacent numbers are inclined away from the common marking. The transposition of the associated numbers and their upward divergence greatly facilitate accurate settings.

It is well understood that in slide rules the slide may be rectilinearly movable in a slot, as herein shown, or in a channel, or the rule may be in disc or cylinder form with a rotary slide member. When in the following claims a slide member is recited, it is to be understood that it may be of any of the known types. Variations from the specific disclosure herein, for example in the relative positioning of scales, are possible and are contemplated in the claims which follow.

I claim:

1. A slide rule comprising a body member and a. slide member, a scale of unit length on said slide member graduated in accordance with the logarithms of numbers from 1 to i0, a cooperating corresponding scale of unit length on said body member graduated in accordance with the logarithms of numbers from 1 to 10, said scales constituting the basic scales of the rule, a scale on said body member graduated in accordance with the logarithms of the logarithms of numbers greater than unity and indexed at the base of the natural iogarithms, a scale on said body member graduated in accordance' with the logarithms of the co-logarithms of numbers less than unity and indexed at the reciprocal of the base of the natural logarithms, the two last-named scales being each composed of at least one section having the same unit length as said basic scales for direct association with said basic scales.

2. A slide rule comprising two body members and a slide member, a scale of unit length on said slide member graduated in accordance with the logarithms of numbers from 1 to 10, a cooperating corresponding scale of unit length on one of said body members graduated in accordance with the logarithms of numbers from 1 to 10, said scales constituting the basic scales of the rule, a iirst series of scales on one of said body members graduated in accordance with the logarithms of the logarithms of numbers greater than unity each successive scale of the series positioned farther away from the basic scales being for successively smaller numbers approaching unity, a

second series of scaleson the other of said body members graduated in accordance with the logarithms of the co-logarithms of positive numbers less than unity each successive scale of the series positioned farther away from the basic scales being for successively larger numbers approaching unity, the two last-named series of scales being each composed of sections having the same unit length as said basic scales for direct association with said basic scales, said first and second series of scales being indexed with respect to the basic scale on the-body member so that graduations on one series of scales are directly opposite graduations representing reciprocal values on the other series of scales.

3. A slide rule comprising a body member and a slide member, a scale of unit length on said slide member graduated in accordance with the logarithme of numbers from 1 to 10, a cooperating corresponding scale of unit length on said body member graduated in accordance with the logarithms of numbers from 1 to 10, said scales constituting the basic scales of the rule, a first series of scales on said body member graduated in accordance with the logarithms of the logarithms of numbers greater than unity, a second series of scales on said body member graduated in accordance with the logarithms of the cologarithms of numbers less than unity, the two last-named series of scales being each composed of sections having the same unit length as said basic scales for direct association with said basic scales, said ilrst and second series of scales being indexed with respect to the basic scale on the body member so that graduations on one series of scales are directly opposite graduations representing reciprocal values on the other series of scales.

4. A slide rule comprising a body member and a slide member, a scale on one of said members graduated in accordance with the logarithms of numbers, a ilrst series of scales on one of said members graduated in accordance with the logarithms of the logarithms of numbers greater than unity of which said scale graduated in accordance with the logarithme of numbers is its basic scale, a rst set of diilerent symbols, each oi the scales of the iirst series of scales being identified by a specic symbol of said first set of symbols, a scale on the other of said members graduated in accordance with the logarithms of numbers, a second series of scales graduated in accordance with the logarithme of the co-logarithms of num- 9 bers less than unity indexed at the reciprocal of the index oi' the scale graduated in accordance with the logarlthms of logarithms of numbers greater than unity and of which the scale on the other of said members graduated in accordprising, a ilrst series of scales on said body memv ance with the logarithms of numbers is its basic ber graduated in accordance with the logarithms scale, a second set of symbols duplicating those of the logarithms of numbers greater than unity, of the first set of symbols, each scale of the seca first set of diierent symbols, each of the scales ond series of scales which is graduated for values of the first series of scales being identified by a reciprocal to a. scale of the first series of scales 10 specic symbol of said first set of symbols, a. secbelng identified by that symbol duplicating that ond series of scales on said body member graduidentifying said scale of the iirst series of scales. ated in accordance with the lagorithms of the co- 5. In a slide rule comprising a body member logarithms of numbers less than unity, the two and a slide member having a scale of unit length last-named series of scales being each composed on said slide member graduated in accordance 15 of sections having the same unit length as saidv with the logarithms of numbers from 1 to 10, basic scales for direct association with said basic a cooperating corresponding scale ot unit length scales, said rst and second series of scales beon said body member graduated in accordance ing indexed with respect to the basic scale on with the logarithms or numbers from 1 to 10, said the body member so that graduatlons on one scales constituting the basic scales of the rule, series of scales are directly opposite graduations and a. series of trigonometric scales giving the representing reciprocal values on the other series values of trigonometric functions on said basic of scales, and a second set of symbols duplicatscales the improvement comprising, a rst series ing those of the tlrst set of symbols, each scale oi' of scales on said body member graduated in acthe second series of scales which is graduated cordance with the logarithms of the logarithms. for values reciprocal to a scale of the rst series of numbers greater than unity, a second series of of Scales being identied by the symbol duplicatscales on said body member graduated in according that identifying said scale of the rst series ance with the logarithms of the co-logarithms of of scales. numbers less than unity, the two last-named JAMES R. BLAND. series of scales being each composed of sections having the same unit length as said basic scales REFERENCES CITED for direct association with Said basic scales Said The following references are of record in the iiritl and second series oi* scales teird indexed me of this patent: w respect to the basic scale on e y member so that graduations on one series of scales are UNITED STATES PATENTS directly opposite graduations representing recip- Number Name Date rocal values on the other series o! scales. 2,170,144 Kells et al. Aug. 22, 1939 6. In a slide rule comprising a body member 2,369,819 Dietzgen Feb. 20, 1945 and a slide member having a scale of unit length 1,632,505 Ritow June 14, 1927 on 1said slide member graduated in accordance 40 2,283,473 Tyler et al. May 19, 1942 Wit the logarithms of numbers from 1 to 10, a cooperating corresponding scale of unit length on FOREIGN PATENTS said body member graduated in accordance with Number y Country Date the logarithms of numbers from 1 to 10, said 18,218 England 1907 Disclaimer 2,422,649.James R. Bland, East rt,

1947. Disclaimer filed Company.

Md. SLIDE RULE. 16, 1948, by the assignee, Keu'el de Esser 10 scales constituting the'basic scales of the rule, and a series of trigonometric scales on the slide member giving the values of trigonometric functions on said basic scales the improvement com- Patent dated June 17,

Hereby enters this disclaimer to claim 3 in the specification. T

[Qfical Gazette January 11, 1949.] 

